Inequalities by P. P. Korovkin
By P. P. Korovkin
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Additional info for Inequalities
12 , Now let us return to the quantity C. We have 1 1 111 1 . 10-4 it will be required to find only 5 roots and to produce a number of arithmetic operations. 539. 55 Problem 2. Calculate the number 1 A==1+ ~/_ 1" 2 1 + ~/_13 + ... +~ J/ 10 12 JI with an accuracy of up to unity. Solution. By virtue of Theorem 8 A== V2 ( 2- t2 r 1 ~/ 1012 + 1 - + _ 112 ~ V2 ( 2-V2 1- -~ 2 10 1 +... 10 12 1 V2 + V3 - ·· · -. 1012 • The first term can be easily found and with a high degree of accuracy by means of the inequalities (28).
Denoting the durability of the beam by P, we get P = kxy2 = kx (d 2 - x 2) = k (d 2x - x 3 ) . The function d2x - x 3 takes the greatest value when 1 d2 x= ( 3 )3='1 = va d ' y2=d2-x2= T2 d2, Y= va d ,j- v 2 =< x V-2. 4 =5. Problem 2. Find the greatest value of the function y sin x sin 2x. == Solution. ~~os x and, hence, -1 ~ z ~ 1. Thefunction z - Z3 = (: "z (1 - Z2) takes a negative value when -1 ~ Z < 0, = ::::E:. ,~ Fig. positive 'value when" 0< z ~1. Therefore, the greatest value of the function is gained in the interval 0 < z ~ '1.
1012 • The first term can be easily found and with a high degree of accuracy by means of the inequalities (28). 10 )4 _ ( \Ql 2)4 1-! f09. 2-,/2 3 4 By virtue of Lemma 1 the sum 1 +_1 _. 10 12 is positive and is not greater than the first term. 109 • The extreme numbers differ from each other by 2, and from the number A by less than 2. 10 9 - 1 differs from A by less than unity. 3 +~, I ~ I < 1. Notice that the accuracy of calculating the number A, containing a trillion of addends, is extremely high.